The Deligne–Mumford compactification of the real multiplication locus and Teichmüller curves in genus 3

By Matt Bainbridge, Martin Möller. 2012

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In the moduli space $$ \mathcal{M} $$g of genus-g Riemann surfaces, consider the locus $$ \mathcal{R}{\mathcal{M}_{\mathcal{O}}} $$ of Riemann surfaces whose Jacobians have real multiplication by the order $$ \mathcal{O} $$ in a totally real number field F of degree g. If g = 3, we compute the closure of $$ \mathcal{R}{\mathcal{M}_{\mathcal{O}}} $$ in the Deligne–Mumford compactification of $$ \mathcal{M} $$g and the closure of the locus of eigenforms over $$ \mathcal{R}{\mathcal{M}_{\mathcal{O}}} $$ in the Deligne–Mumford compactification of the moduli space of holomorphic 1-forms. For higher genera, we give strong necessary conditions for a stable curve to be in the boundary of $$ \mathcal{R}{\mathcal{M}_{\mathcal{O}}} $$. Boundary strata of $$ \mathcal{R}{\mathcal{M}_{\mathcal{O}}} $$ are parameterized by configurations of elements of the field F satisfying a strong geometry of numbers type restriction.We apply this computation to give evidence for the conjecture that there are only finitely many algebraically primitive Teichmüller curves in $$ \mathcal{M} $$3. In particular, we prove that there are only finitely many algebraically primitive Teichmüller curves generated by a 1-form having two zeros of order 3 and 1. We also present the results of a computer search for algebraically primitive Teichmüller curves generated by a 1-form having a single zero.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/fa34a4609e22cdeb5ea02e0c378cee733c0a90c2